Neue Lösungsstrategien für l1-Minimierungsprobleme mit Kalman-Filtern

In many problems, high-dimensional discrete signals need to be reconstructed from
noisy and often undersampled data, raising the issue of solving nominally underdetermined
noise-contaminated systems of equations. The theory of compressed sensing
states (and proves) that such signals can in fact uniquely be reconstructed under
the sparsity assumption. In the broad research field of compressed sensing – whose
applications are found, for example, in medicine (MRT), radar and high-frequency
communication technology, many algorithms already exist for the reconstruction of
sparse signals. One of the most important properties is the so-called null space property
of the sensor matrix. It ensures that the sparse or compressible representation
can be recovered by the ℓ1-minimization, which can in fact be realized either by convex
optimization approaches, which is the classical way, or alternatively by estimationtheoretic
approaches, e.g. by extended linearized Kalman-filters, which is the approach
analyzed in this paper.
In this work new results for the reconstruction of sparse signals by ℓ1-minimization
are established with such a Kalman-filter. The core of our Kalman-filter is based on a
simple idea. On the basis of a particular solution x_p (there are infinitely many solutions),
the Kalman-filter estimates another solution from the null space room of the
measurement matrix in each iteration step, so that the sum of both vectors achieves
a solution x = x_p + x_N with a reduced ℓ1-norm. In the first part, the Kalman-filter
uses convergence acceleration techniques to construct a convergent sequence whose
limit value provides a solution, which corresponds to Chambolle & Pock’s primal-dual
algorithm solution. The convergence acceleration methods result from the Delat2-basic
process of Aitken.
For solving ℓ1-minimization problems so-called thresholding methods are increasingly
used to find the solution. In the second part of the thesis the Kalman-filter is presented
with an external thresholding method. With this external thresholding, which does not
directly affect the Kalman-filter, we can now reconstruct sparse signals very quickly.
Further investigations of noise-affected signals with the modified Kalman-filter confirm
the results in terms of sparse recovery error, ℓ0-norm of the estimates, support mismatch,
and recovery time compared to the usual known ℓ1-minimization algorithms,
e.g., primal-dual algorithm of Chambolle & Pock and Orthogonal Matching Pursuit.